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Mirrors > Home > LLPE Home > Th. List > md1 | Structured version |
Description: ⊥ is the unit of ⅋ (left side). |
Ref | Expression |
---|---|
md1 | ⊦ (𝜑 ⧟ (⊥ ⅋ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-init 7 | . . . . . . 7 ⊦ (~ 𝜑 ⅋ 𝜑) | |
2 | 1 | ax-ibot 4 | . . . . . 6 ⊦ (⊥ ⅋ (~ 𝜑 ⅋ 𝜑)) |
3 | 2 | mdcoi 12 | . . . . 5 ⊦ ((~ 𝜑 ⅋ 𝜑) ⅋ ⊥) |
4 | 3 | mdasi 14 | . . . 4 ⊦ (~ 𝜑 ⅋ (𝜑 ⅋ ⊥)) |
5 | 4 | mdcod 11 | . . 3 ⊦ (~ 𝜑 ⅋ (⊥ ⅋ 𝜑)) |
6 | 5 | dfli2i 66 | . 2 ⊦ (𝜑 ⊸ (⊥ ⅋ 𝜑)) |
7 | ax-init 7 | . . . . . 6 ⊦ (~ (⊥ ⅋ 𝜑) ⅋ (⊥ ⅋ 𝜑)) | |
8 | 7 | mdcod 11 | . . . . 5 ⊦ (~ (⊥ ⅋ 𝜑) ⅋ (𝜑 ⅋ ⊥)) |
9 | 8 | mdasri 17 | . . . 4 ⊦ ((~ (⊥ ⅋ 𝜑) ⅋ 𝜑) ⅋ ⊥) |
10 | 9 | ebotr 19 | . . 3 ⊦ (~ (⊥ ⅋ 𝜑) ⅋ 𝜑) |
11 | 10 | dfli2i 66 | . 2 ⊦ ((⊥ ⅋ 𝜑) ⊸ 𝜑) |
12 | 6, 11 | ilb 96 | 1 ⊦ (𝜑 ⧟ (⊥ ⅋ 𝜑)) |
Colors of variables: wff var nilad |
Syntax hints: ⊥wbot 1 ⅋ wmd 2 ~ wneg 3 ⧟ wlb 55 |
This theorem was proved from axioms: ax-ibot 4 ax-ebot 5 ax-cut 6 ax-init 7 ax-mdco 8 ax-mdas 9 ax-iac 32 ax-eac1 33 ax-eac2 34 |
This theorem depends on definitions: df-lb 56 df-li 62 |
This theorem is referenced by: md2 114 |
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